Polymer Rheology by Dielectric Spectroscopy · 2018. 9. 25. · 0 Polymer Rheology by Dielectric Spectroscopy Clement Riedel 1, Angel Alegria 1, Juan Colmenero 1 and Phillipe Tordjeman

0

Polymer Rheology by DielectricSpectroscopy

Clement Riedel1, Angel Alegria1, Juan Colmenero1

and Phillipe Tordjeman2

1Universidad del Pais Vasco / Euskal Herriko Unibertsitatea2Université de Toulouse, INPT

1Spain2France

1. Introduction

The aim of this chapter is to discuss the main models describing the polymer dynamics atmacroscopic scale and present some of the experimental techniques that permit to test thesemodels. We will notably show how Broadband Dielectric Spectroscopy (BDS) permits toobtain rheological information about polymers, i.e. permits to understand how the matterflows and moves. We will focus our study the whole chain motion of cis-polyisoprene 1,4 (PI).

Due to dipolar components both parallel and perpendicular to the chain backbone, PIexhibits a whole chain dielectric relaxation (normal mode) in addition to that associated withsegmental motion. The Rouse model (Rouse, 1953), developed in the fifties, is well knownto describe rather correctly the whole chain dynamics of unentangled polymers. In the past,the validity of the Rouse model has ever been instigated by means of different experimentaltechniques (BDS, rheology, neutron scattering) and also by molecular dynamics simulations.However this study is still challenging because in unentangled polymer the segmentaldynamics contributions overlap significantly with the whole chain dynamics. In this chapter,we will demonstrate how we have been able to decorelate the effect of the α-relaxation on thenormal mode in order to test how the Rouse model can quantitatively describes the normalmode measured by BDS and rheology. The introduction of polydispersity is a key point of thisstudy.

The reptational tube theory, first introduced by de Gennes de Gennes (1979) and developedby Doi and Edwads Doi & Edwards (1988) describes the dynamics of entangled polymerswhere the Rouse model can not be applied. Many corrections (as Contour Length Fluctuationor Constraints Release) have been added to the pure reptation in order to reach a totallypredictive theory McLeish (2002); Viovy et al. (2002). In the last part of this chapter, we willshow how the relaxation time of the large chain dynamics depends on molecular weight anddiscuss the effects of entanglement predicted by de Gennes theory on dielectric spectra. PIis a canonical polymer to study the large chain dynamics; however, due to its very weakrelaxation, the measurement of this dynamics at nanoscale is still challenging.

1

www.intechopen.com

2 Will-be-set-by-IN-TECH

2. Material and methods

2.1 Polyisoprene samples

Polymers are characterized by a distribution of sizes known as polydispersity. Given adistribution of molecular sizes (Ni molecules of mass Mi), the number-averaged molecularweight Mn is defined as the first moment of distribution (Eq. 1) and the mass-averagedmolecular weight Mw as the ratio between the second and first moments of the distribution(Eq. 2).

Mn = ∑i

Ni Mi

Ni(1)

Mw = ∑i

Ni M2i

Ni Mi(2)

The polydispersity Ip is the ratio Mw/Mn.

Polymers can have dipoles in the monomeric unit that can be decomposed in two differentcomponents: parallel or perpendicular to the chain backbone. The dipole moment parallelto the chain backbone giving rise to an "end-to-end" net polarization vector will induce theso-called dielectric normal mode dielectric relaxation that can be studied using theoreticalmodels. The dipole moment perpendicular to the chain backbone will lead to the segmentalα-relaxation that can only be described using empirical models, since no definitive theoriticalframework exists for this universal process.

We have chosen to study the relaxations of two different samples. The first is1,4-cis-polyisoprene (PI). This isomer is a A-type polymer in the Stockmayer classificationStockmayer (1967): it carries both local dipole moments parallel and perpendicular to thechain backbone. The chapter on the large scale dynamics will be focused on the study of thedynamics of the normal mode of PI.

The polyisoprene (PI) samples were obtained from anionic polymerization of isoprene (Fig.1). According to the supplier, Polymer Source, the sample is linear (no ramification) and itsmicro-composition is 80% cis, 15% trans and 5% other 1. We are working with the isomer cisthat have a net end-to-end polarization vector. Before the experiments samples were dried ina vacuum oven at 70 žC for 24 hours to remove any trace of solvent.

(a) Polymerization of isoprene (b) Isomer cis and trans

Fig. 1. Polymerization and isomer of isoprene

For our study, seven samples have been chosen to cover a large band of molecular weightwith low polydispersity (Table 1). The polydispersity (determined from size-exclusionchromatography experiments) is given by supplier while the glass transition temperature (Tg)has been measured by differential scanning calorimetry. To avoid oxidation, PI samples werestored at -25 žC.

4 Rheology

www.intechopen.com

Polymer Rheology by Dielectric

Spectroscopy 3

Sample Mn [kg/mol] Mw [kg/mol] Ip Tg [K]

PI-1 1.1 1.2 1.11 194

PI-3 2.7 2.9 1.06 203

PI-10 10.1 10.5 1.04 209

PI-33 33.5 34.5 1.04 210

PI-82 76.5 82 1.07 210

PI-145 138 145 1.07 210

PI-320 281 320 1.14 210

Table 1. Molecular weight, polydispersity and Tg of PI

2.2 Rheology

Elasticity is the ability of a material to store deformational energy, and can be viewed as thecapacity of a material to regain its original shape after being deformed. Viscosity is a measureof the ability of a material to resist flow, and reflects dissipation of deformational energythrough flow. Material will respond to an applied force by exhibiting either elastic or viscousbehavior, or more commonly, a combination of both mechanisms. The combined behavior istermed viscoelasticity. In rheological measurements, the deformational force is expressed asthe stress, or force per unit area. The degree of deformation applied to a material is called thestrain. Strain may also be expressed as sample displacement (after deformation) relative topre-deformation sample dimensions.

Dynamic mechanical testing involves the application of an oscillatory strain γ(t) = γ0 cos(ωt)to a sample. The resulting sinusoidal stress σ(t) = σ0 cos(ωt + δ) is measured andcorrelated against the input strain, and the viscous and elastic properties of the sample aresimultaneously measured.

If the sample behaves as an ideal elastic solid, then the resulting stress is proportional to thestrain amplitude (Hooke’s Law), and the stress and strain signals are in phase. The coefficientof proportionality is called the shear modulus G. σ(t) = G γ0 cos(ωt) If the sample behavesas an ideal fluid, then the stress is proportional to the strain rate, or the first derivative of thestrain (Newton’s Law). In this case, the stress signal is out of phase with the strain, leading itby 90r. The coefficient of proportionality is the viscosity η. σ(t) = η ω γ0 cos(ωt + π/2)

For viscoelastic materials, the phase angle shift (δ) between stress and strain occurssomewhere between the elastic and viscous extremes. The stress signal generated by aviscoelastic material can be separated into two components: an elastic stress (σ’ ) that is inphase with strain, and a viscous stress (σ” ) that is in phase with the strain rate (dγ/dt) but90r out of phase with strain. The elastic and viscous stresses are sometimes referred to asthe in-phase and out-of-phase stresses, respectively. The elastic stress is a measure of thedegree to which the material behaves as an elastic solid. The viscous stress is a measureof the degree to which the material behaves as an ideal fluid. By separating the stress intothese components, both strain amplitude and strain rate dependence of a material can besimultaneously measured. We can resume this paragraph by a set of equation:

σ(t) =

(

σ0 cos(δ)

γ0

)

γ0 cos(ω t) +

(

σ0 sin(δ)

γ0

)

γ0 sin(ω t) (3)

5Polymer Rheology by Dielectric Spectroscopy

www.intechopen.com


G′ =σ0 cos(δ)

γ0G′′ =

σ0 cos(δ)

γ0(4)

G(ω) = G′ + iG′′ (5)

2.3 Broadband dielectric spectroscopy

The set-up of a broadband dielectric spectroscopy (BDS) experiment is displayed in Fig. 2. Thesample is placed between two electrodes of a capacitor and polarized by a sinusoidal voltageU(ω). The result of this phenomena is the orientation of dipoles which will create a capacitivesystem. The current I(ω) due to the polarization is then measured between the electrode. Thecomplex capacity C(ω) of this system is describe by the complex dielectric function ǫ(ω) as:

ǫ(ω) = ǫ′(ω)− i ǫ′′(ω) =C(ω)

C0=

J(ω)

i ω ǫ0E(ω)=

I(ω)

i ω U(ω)C0(6)

where: C0 is the vacuum capacitance of the arrangement

E(ω) is the sinusoidal electric field applied (within the linear response)J(ω) is the complex current density ǫ0 = 8.85 10−12 A s V−1 m−1is the permittivity of vacuum

ǫ′ and ǫ′′ are proportional to the energy stored and lost in the sample, respectively.

(a) Schema of the principle of BDS (b) Polarization of matterand resulting capacitivesystem

Fig. 2. Schema of the principle of BDS and polaryzation of matter

In our experiments samples were placed between parallel gold-plated electrodes of 20 mmdiameter and the value of the gap (between the electrodes) was fixed to 0.1mm (by a narrowPTFE cross shape piece).

Polarization of matter (−→P ) can be describe as damped harmonic oscillator. When the

electromagnetic field is sinusoidal the dipole oscillates around its position of equilibrium.This response is characterized by ǫ(ω):

−→P = (ǫ(ω)− 1)ǫ0

−→E (ω) (7)

The complex dielectric function is the one-sided Fourier or pure imaginary Laplace

transform of the correlation function of the polarization fluctuations φ(t) = [−→P (t) · −→P (0)]

(phenomological theory of dielectric relaxation Kremer & Schonals (2003))

ǫ(ω)− ǫ∞

ǫs − ǫ∞=

∫ ∞

0exp(−i ω t)

(

− dφ(t)

dt

)

dt (8)

6 Rheology

www.intechopen.com


Spectroscopy 5

where ǫs and ǫ∞ are are the unrelaxed and relaxed values of the dielectric constant.

3. Resolving the normal mode from the α-relaxation

For low molecular weight, the normal mode of PI overlaps at high frequency with theα-relaxation (Fig. 3). This overlapping is an intrinsic problem in checking the Rouse

Fig. 3. Dielectric relaxation curves collected at 250 K on PI with different molecular weights.

model (that describes the normal mode dynamics, see next section), since its applicabilityis limited to chains with moderate molecular weight (below the molecular weight betweenentanglements). In fact, even by microscopic techniques with spatial resolution as neutronscattering, it is rather difficult to distinguish the border between chain and segmentalrelaxations Richter et al. (2005).Because the time scale separation between the two dynamicalprocesses is not complete, a detailed analysis of the validity of the Rouse model predictions athigh frequencies requires accounting accurately for the α-relaxation contribution.

The imaginary part of the dielectric α-relaxation can be analyzed by using thephenomenological Havriliak-Negami function:

ǫ(ω) = ǫ∞ +ǫs − ǫ∞

(1 + (i ω τHN)α)γ. (9)

In addition, a power law contribution (∝ ω−1) was used to account for the normal modecontribution at low frequencies, which is the frequency dependence expected from the Rousemodel for frequencies larger than the characteristic one of the shortest mode contribution.Thus, we assumed that the high frequency tail of the normal mode follows a C/ω law andsuperimposes on the low frequency part of the alpha relaxation losses, being C a free fittingparameter at this stage. The α-relaxation time corresponding to the loss peak maximum wasobtained from the parameters of the HN function as follows Kremer & Schonals (2003):

τα = τHN

[

sin(

α γπ2+2γ

)]1/α

[

sin(

α π2+2γ

)]1/α(10)


www.intechopen.com


The results obtained for samples with different molecular weight at a common temperature of230 K are shown in Fig. 4. It is evident that in the low molecular weight range the time scale ofthe segmental dynamics is considerably faster than that corresponding to the high molecularweight limit. This behavior is mainly associated with the plasticizer-like effect of the enchaingroups, which are more and more relevant as chain molecular weight decreases. The line infigure 4 describing the experimental behavior for the PI samples investigated is given by:

τα(M) =τα(∞)

(1 + 10000/M)2(11)

As can be seen in Fig. 3 and 4, the α-relaxation contribution from long chains is nearly

Fig. 4. α-relaxation times of polyisoprene at 230 K as a function of molecular weight. The linecorresponds to Eq. 11

independent on molecular mass, but for low molecular masses (c.a. below 20 000 g/mol)it is shifted to higher frequencies and slightly broader. Because the rather small length scaleinvolved in the segmental dynamics (around a nanometer) Cangialosi et al. (2007); Inoue et al.(2002a) it is expected that for a relatively low molecular weight sample, as PI-3, there wouldbe contributions to the α-relaxation with different time scales originated because the presenceof chains of different lengths. This could explain the fact that the α-relaxation peak of the PI-3sample is slightly broader than that of a high molecular weight one, the PI-82 for instance(see Figure 3). Thus, in order to take this small effect into account we decided to describe theα-relaxation of the PI-3 sample as a superposition of different contributions. The contributionto the α-relaxation from a single chain of molecular weight M is assumed to be of the HNtype (Eq. 9), with intensity proportional to the number of units (segments) in the chain, i.e.proportional to M. Under this assumption, the whole α-relaxation would be given as follows:

ǫα(ω) =∫

∆ǫ(M)

(1 + (i ω τHN)β)γg(M)dM (12)

=∆ǫα

MN

∫

M

(1 + (i ω τHN)β)γg(M)dM (13)

8 Rheology

www.intechopen.com


Spectroscopy 7

where g(M) is a Gaussian-like distribution introduced to take into account the effects of theactual molecular weight distribution of the sample:

g(M) =1√

2 πσexp

(

− (M − Mn)2

2σ2

)

(14)

σ = Mn

√

Mw

Mn− 1 (15)

g(M)dM is the number density of chains with molecular weight M, ∆ǫα is the total dielectricstrength associated to the α-relaxation, and 1/Mn is just a normalization factor. Theparameters β and γ in Eq. 13 were taken form the fitting of the α-relaxation losses of a highmolecular weight sample, PI-82, (β = 0.71, γ = 0.50), i.e. the shape of each component hasbeen assumed to be that obtained from the experiment in a sample with a high molecularweight and a narrow distribution. For such a sample no differences between the contributionsof distinct chains are expected (see Figure 4). Furthermore, note that for this sample theα-relaxation is very well resolved from the normal mode and, therefore, its shape can beaccurately characterized. Moreover, according to Eq. 11, the following expression for τHN(M)was used, τHN(M)/τHN(∞) = [1 + (10000/M)2]−1, where a value of τHN(∞) = τHN(82000)is a good approximation. In order to avoid the unphysical asymptotic behavior (ǫ′′ ∝ ωβ)given by the HN equation at very low frequencies, the physical asymptotic behavior (ǫ′′ ∝

ω) was imposed for frequencies two decades lower than that of the peak loss frequency.This cut-off frequency was chosen because is the highest cut-off frequency allowing a gooddescription of the α-relaxation data from the high molecular weight PI samples. The resulting

Fig. 5. Resolved normal of PI-3, the dotted line represents the modelled contribution of theα-relaxation. Reprinted with permsision from Riedel et al, Macromolecules 42, 8492-8499Copyright 2009 American Chemical Society

curve is depicted in Fig. 5 as a dotted line. The value of ∆ǫα (single adjustable parameter) hasbeen selected to fit the experimental data above f=5e4 Hz, where no appreciable contributionsfrom the normal mode would be expected. After calculating the α-relaxation contribution, the


www.intechopen.com


normal mode contribution can be completely resolved by subtracting it from the experimentaldata. The so obtained results are depicted in Figure 5 for PI-3.

4. Rouse model: experimental test

The large chain-dynamics of linear polymers is one of the basic and classical problemsof polymer physics, and thereby, it has been the subject of intensive investigation, bothexperimentally and theoretically, over many years Adachi & Kotak (1993); de Gennes (1979);Doi & Edwards (1988); Doxastakis et al. (2003); Hiroshi (2001); McLeish (2002); Watanabeet al. (2002a); Yasuo et al. (1988). Despite of the broad range of models and theoreticalapproaches existing in the literature there are many aspects of the problem that remainto be understood (see refs.Doi & Edwards (1988); Doxastakis et al. (2003); McLeish (2002);Rouse (1953) and references therein). Most of the current investigations are devoted to theproblem of the dynamics of highly entangled polymer melts with different architectures andtopologies, and to the rheology of polymer systems of industrial relevance Doi & Edwards(1988); Likhtman & McLeish (2002); Watanabe et al. (n.d.). Concerning the chain-dynamicsof unentangled polymers, it is generally assumed that the well-known Rouse model Rouse(1953) provides a suitable theoretical description. The Rouse model represents a linear chainas a series of beds and springs subjected to entropic forces in a medium with a constantfriction. Although this simple approach obviously fails in describing the melt dynamics oflong chains at longer times, the Rouse model is also used for describing the fastest part of theresponse of these long chains and thereby it is a common ingredient of all available modeland theories. In the past, the validity of the Rouse model has ever been instigated by means ofdifferent experimental techniques and also by molecular dynamics simulations. But as alreadymentioned in the previous section, a full and detailed test of the Rouse model is challengingbecause in unentangled polymer melts the segmental dynamics (α-relaxation) contributionsoverlap significantly with the high-frequency components of the chain dynamics. This fact,among others, restricts the use of rheology experiments to test accurately the Rouse model onunentangled polymer chains. It is very hard to obtain rheology data of unentanged polymersin the melt. This is due to the rapid relaxation times of the material and the broad spread of theeffect of more local molecular mechanisms that affect the stress relaxation modulus at higherfrequencies. We will first detail how we have been able to resolve the normal mode from theα-relaxation in BDS data. Then, we will start our study of the Rouse model by introducing theexpression of both complex dielectric permittivity and shear modulus before presenting thefull detail of the test of the Rouse model using both rheology and BDS.

4.1 Theory

The random walk is an extremely simple model that accounts quantitatively for manyproperties of chains and provides the starting point for much of the physics of polymer. In thismodel, we consider an ideal freely joined chain, made up of N links, each define by a vector−→r i and b the average length between the links (Fig. 6). The different links have independentorientations. Thus the path of the polymer in space is a random walk.

The end-to-end vector−→R N(t) is the sum of the N jump vectors −→r i+1 −−→r i which represent

the direction and size of each link in the chain:

−→R N(t) =

N−1

∑i=1

−→r i+1 −−→r i (16)

10 Rheology

www.intechopen.com


Spectroscopy 9

Fig. 6. Schematic representation of the spring model for polymer chain

The mean end-to-end distance is

〈−→R N(t)2〉 = N b2 (17)

The overall size of a random walk is proportional to the square root of the number of steps.

The correlation function of−→R N(t) can be expressed as the sum of the vibration of the so-called

Rouse modes:

〈−→R N(t)−→R N(0)〉 = 2b2

N

N−1

∑p:odd

cot2( pπ

2N

)

exp

(−t

τp

)

(18)

with

τp =ξb2

12 kb T sin2( p π

2N

) (19)

In the limit of the long chains, for pπ<2N, we obtain the well known expression of theend-to-end vector:

〈−→R N(t)−→R N(0)〉 ∼= 8 b2N

π2 ∑p:odd

1

p2exp

(

− t

τp

)

(20)

Due to the factor 1/p2 the correlation function of the end-to-end vector is dominated by theslowest mode and as τp = τ1

p2 the higher modes are shifted to higher frequencies.

For small deformation, the chain can be approximatively described using a gaussianconfiguration and the expression of the xy component of the shear stress is given by Doi &Edwards (1988):

σxy =3 ν kbT

Nb2

N

∑i=1

〈(−→r i+1 −−→r i)x(−→r i+1 −−→r i)y〉 (21)

where ν is the number of chain per volume unit.


www.intechopen.com


4.1.1 Expression of ǫ(ω) in the frame of the Rouse model

BDS measure the time fluctuation of the polarization. Polymers can have dipole momentsparallel to the chain backbone, leading to a net "end-to-end" vector. The part of thepolarization related with these dipole moments is proportional to the time fluctuation of the

end-to-end vector: φ(t) = [−→P (t) · −→P (0)] ∝ [

−→RN(t) · −→RN(0)]. Using the expression of the

end-to-end vector (Eq. 18) in the phenomenological theory of the dielectric relaxation (Eq.8) we obtain an expression of the frequency dependance of the dielectric permittivity in theframe of the Rouse model:

ǫ′(ω) ∝2b2

N

N−1

∑p:odd

cot2( pπ

2N

) 1

1 + ω2τ2p

(22)

ǫ′′(ω) ∝2b2

N

N−1

∑p:odd

cot2( pπ

2N

) ωτp

1 + ω2τ2p

(23)

The factor cot2(p) (which can be developed in 1/p2 when p<N) strongly suppress thecontribution of high p modes. Therefore, the dielectric permittivity is sensitive to slow (low pmodes) and facilitates resolving normal mode and segmental relaxation contributions.

4.1.2 Expression of G(ω) in the frame of the Rouse model

We can calculate the expression of the shear modulus in the time domain from Eq.21, andusing a Fourier transform, we obtain:

G′(ω) ∝N−1

∑p=1

ω2τ2p /4

1 + ω2τ2p /4

(24)

G′′(ω) ∝N−1

∑p=1

ωτp/2

1 + ω2τ2p /4

(25)

All the modes are contributing in the expression of the shear modulus. Therefore, in the case ofsmall chains, rheology is not a convenient experimental technique to resolve the chain modesfrom the segmental dynamics.

4.2 Test of the Rouse model

4.2.1 Choice of the sample: Working below ME

The Rouse model describe the dynamics of unentangled polymers only. Therefore it isimportant to characterize its domains of application. As we will see below, the use ofthe time temperature superposition (TTS, see next paragraph) to create master curves inrheology is questionable. Figure 7 represents the master plot obtained for PI-82 from differenttemperatures T=[40, 20, -15, -30, -50]žC (each represented by a different color) shifted tothe reference temperature of -50žC. Master curves permits to obtain the full rheology of thepolymer. At low frequencies we can see the Maxwell zone (G′ ∝ ω,G′′ ∝ ω2). Then, theelastic plateau, where G’>G”, is well defined and the polymer is therefore entangled. Afterthe plateau, the Rouse zone is characterized by G′ ∝ G′′ ∝ ω1/2. Rheology allows measuringa value of the molecular weight of entanglement ME by measuring G0

n, value of G’ at the

12 Rheology

www.intechopen.com


Spectroscopy 11

Fig. 7. Master curve of PI-82 at the reference temperature of -50žC

minimum of tan(δ) (around the middle of the elastic plateau). This measurement is made atone temperature, and therefore no master curve is needed to get the value of G0

n. Accordingto reference Doi & Edwards (1988):

ME =ρRT

G0n

(26)

where ρ=0.9 g/cm3 is the density, R = 8.32 J/K/mol is the ideal gas constant and T is thetemperature of measurement. Using the value obtained from Fig. 7: G0

n = 2.105 Pa, we obtaina value of ME=9 kg/mol.

The dielectric normal mode reflects the fluctuations of the end-to-end vector and is dominatedby the slowest chain normal mode. As higher modes are scaled and shifted to higherfrequencies, the timescale of the normal mode peak, τN = 2π

fN(where fN is the frequency

of the maximum of the peak), measured by BDS provides a rather direct access to the Rousetime τ1 when Rouse theory is fulfilled. Using these values of τN at 230 K we have checkedwhether below the molecular weight between entanglements the Rouse model predictionsconcerning the molecular weight dependence of the slowest relaxation times (τ1 ∝ M2) isverified. In Fig. 8 we present the values of the ratio τN/τα as a function of the molecularmass. The ratio between the longest relaxation time and the value of τα obtained at thesame temperature in the same experiment is the way used trying to remove the possiblevariation in τN arising from the monomeric friction coefficient, which can be assumed asstraightforwardly related with the changes in the glass transition temperature, and hence,with the noticeable effect of end chain groups in the segmental dynamics. Fig. 8 shows thatbelow a molecular weight of around 7 000 g/mol the data scales approximately with M2. Thisis just the Rouse model prediction, i.e., what is deduced from Eq. ?? for low-p values wherethe following approximation holds. This value is intermediate between that of the molecularmass between entanglements (Me=9 kg/mol) obtained by rheology and that measured fromneutron scattering experiments Me=5 kg/mol Fetters et al. (1994).

Once we have confirmed the range of molecular masses where the molecular weightdependence of the longest relaxation time verifies Eq. ??, we will test if the whole dielectric


www.intechopen.com


Fig. 8. Ratio τN/τα , as a function of the molecular weight at 230 K. Solid line represents thebehavior predicted by the Rouse model where τN = τ1 ∝ M2. Reprinted with permsisionfrom Riedel et al, Macromolecules 42, 8492-8499 Copyright 2009 American Chemical Society

normal mode conforms the Rouse model predictions. We have selected the sample PI-3 for thistest because, on the one hand, it has a molecular weight sufficiently below of the molecularweight between entanglements (so all the molecular masses of the distribution are belowME and, on the other hand, it shows a normal mode that is rather well resolved from thesegmental α-relaxation contributions to the dielectric losses. Using higher molecular weightsamples yield the possibility that the high molecular weight tail of the distribution was abovethe entanglement molecular weight. On the contrary, for lower molecular weight samples thestronger superposition of the normal mode and the α-relaxation will make the comparisonless conclusive (see Fig. 3). For the test, we have taken the data recorded at a temperature of230 K where the normal mode contribution is completely included within the experimentalfrequency window, being at the same time the α-relaxation contribution also well capturedand then subtracted (see previous section and Figure 5).

4.2.2 Time temperature superposition (TTS)

Figure 9 a shows the high level of accuracy obtained by means of the present BDS experimentswhen measuring the rather weak dielectric relaxation of a PI-3 sample, for both the normalmode and the α-relaxation. From simple inspection of the data it is apparent that the normalmode peak shifts by changing temperature without any significant change in shape (timetemperature superposition is verified for this process), which is one of the predictions of theRouse model (see eq. 18). However, it is also evident that the shift of the normal mode peak isdistinct than that of the α-relaxation one, i.e. the TTS fails for the complete response. ? This factis illustrated in Figure 9 b where data at two temperatures where both processes are clearlyvisible in the frequency window are compared. For this comparison the axes were scaled(by a multiplying factor) in such a way that the normal peaks superimpose. Whereas thesuperposition in the normal mode range is excellent, the same shift makes the alpha relaxationpeak positions to be about half a decade different. The temperature dependance of the normalmode peak can be described by Williams-Landel-Ferry (WLF) Williams et al. (1995) equation:

log(aT) =C1(T − Tre f )

T − Tre f + C2(27)

When considering the temperature dependence of the shift factor we found that, whereas forhigh molecular masses (above that between entanglements) the WLF parameters are the same

14 Rheology

www.intechopen.com


Spectroscopy 13

(a) Dielectric spectra of PI3 from 220K (left)to 300K (right) ∆T = 10K.

(b) Comparison of normal mode andα-relaxation frequency shifts for PI3 whentemperature is changed from 220 K to 250 K.

Fig. 9. Shift of the dielectric response of PI-3 with temperature. Reprinted with permsisionfrom Riedel et al, Macromolecules 42, 8492-8499 Copyright 2009 American Chemical Society

within uncertainties (WLF parameters with Tg as the reference temperature C1 = 30.20.7 andC2 = 57.00.2K), for lower molecular weight samples the value of C1 remains the same but C2becomes noticeably smaller, being 49.2 K for PI-3. This is likely related with the significantvariation of Tg in the low molecular range (see Table 1).

The data resulting from the rheology experiments performed on a PI-3 sample of 1.3 mmthickness using two parallel plates of 8 mm diameter are shown in Figure 10. To produce thisplot a master curve at a reference temperature Tre f = 230 K was built imposing the horizontalshift factor aT determined from BDS (see above) and a vertical shift factor bT=Tr/T. Curveshave been measured at 215, 220, 225, 230 and 240K, each color represent a temperature. It isapparent that the superposition so obtained is good in the terminal relaxation range, althoughat the highest frequencies, where the contributions of the segmental dynamics are prominent,the data superposition fails clearly. The use of master curves in rheology experiments is astandard practice because of the rather limited frequency range and the general applicabilityof the TTS principle to the terminal relaxation range. However, the presence of the segmentaldynamics contribution at high frequencies when approaching Tg could make the applicationof TTS questionable because the different temperature shifts of global and the segmentaldynamics (see above). Thus, using rheology data alone the high frequency side of the terminalregion could be highly distorted.

4.2.3 Determination of the bead size

To use Eq. 23 we need the previous determination of N, which is not known a priori, as itrequires the estimation of the bead size. Adachi and co-workers Adachi et al. (1990) estimatedthe bead size of PI on the basis of an analysis of the segmental relaxation in terms of adistribution of Debye relaxation times. These authors suggested that a PI bead would containabout 7 monomers. However, a generally accepted approach is to consider that the bead sizewould be of the order of the Kuhn length, bk. Inoue et al. (2002b) The literature value of bk for


www.intechopen.com


Fig. 10. Rheological master curve of PI-3(reference temperature Tr =230 K) of the real: � andimaginary: • parts of the shear modulus.

PI is 0.84 nm,Rubinstein & Colby (2003) which corresponds to a molecular mass of the Kuhnsegment of 120 g/mol. This means that a Kuhn segment would contain about 1.5 monomers,about a factor of about 5 less than Adachi et al. estimated. On the other hand, recent resultshas evidenced that the α-relaxation in the glass transition range probes the polymer segmentalmotions in a volume comparable to b3

k . Cangialosi et al. (2007); Lodge & McLeish (2000) Thus,we will introduce a "segmental" Rouse time τS = τp=N , which apparently have not a clearphysical meaning (the fastest Rouse time would be p=N-1) but it can be related with the

so-called characteristic Rouse frequency (W = 3kbT/b2) as τ−1S = 4W. Nevertheless, τS is

here used as a convenient parameter for the further analysis. We can identify τS at Tg withthe α-relaxation time determined at this same temperature, i.e. τS(Tg)=5 s. In this way, byusing the WLF equation describing the temperature dependence of the normal mode peak,the obtained value of τS at 230 K would be 1.1e-4 s. Using this value and that obtained forτ1 from the normal peak maximum in Eq. ?? a value of N=24 results. Note that accordingwith this value the bead mass would be around 121 g/mol in very good agreement withliterature results for the Kuhn segment mass,Rubinstein & Colby (2003) and consequently thecorresponding bead size will be nearly identical to bk.

4.2.4 Monodisperse system

Thus, in order to test the validity of the Rouse model in describing the dielectric normal moderelaxation data we assumed N=24 in Eq. 23, which would be completely predictive, except inan amplitude factor, once the slowest relaxation time τ1 was determined from the loss peakmaximum. The so calculated curve is shown as a dashed line in Figure ??. It is very clearthat the calculated curve is significantly narrower than the experimental data, not only at highfrequencies where some overlapping contribution from the α-relaxation could exists, but moreimportantly also in the low frequency flank of the loss peak. This comparison evidence clearlythat the experimental peak is distinctly broader than the calculation of the Rouse model basedin Eq. 7, confirming what was already envisaged in Figure 7a of ref. Doxastakis et al. (2003).Nevertheless, an obvious reason for this discrepancy could be the fact that the actual samplehas some (small) polydispersity. Despite of the fact that PI samples with a low polydispersity

16 Rheology

www.intechopen.com


Spectroscopy 15

(1) were chosen, even samples obtained from a very controlled chemistry contains a narrowdistribution of the molecular weights, which was not considered in the previous calculation.

4.2.5 Introduction of polydispersity

Now, we are in a situation where it becomes possible to perform a detailed comparisonbetween the experimental normal mode relaxation and the calculated Rouse modelexpectation. In order to incorporate the small sample polydispersity, the response expectedfrom the Rouse model has been calculated as a weighted superposition of the responsescorresponding to chains with different molecular weights. Since the molecular masses of thechains in the PI-3 sample are all below the molecular weight between entanglements, themolecular weight dependence of τp will be that given by Eq. 19 for all of the different chains.For calculating τp(M) we have used in Eq. 19 a common value τs irrespective of the molecularweight of the particular chain. Furthermore, we calculated τ1(Mw) as the reciprocal of thepeak angular frequency of the experimental normal mode of this sample.

In this way the Rouse model remains completely predictive and the corresponding dielectricresponse to be compared with normal mode contribution from the actual sample will be givenby:

ǫ′′N(ω) =∆ǫN

Mn

∫

M2b2

N

N−1

∑p:odd

cot2( pπ

2N

)

g(M)dM (28)

where the contribution to the normal mode from a given chain is proportional to the chaindipole moment (to the end-to-end vector), and thus again proportional to M.

As can be seen in Figure 11 (solid line), the sum over all the modes and all the molecularweight provides a satisfactory description of the experimental dielectric losses, namely atfrequencies around and above the peak. The excellent agreement evidences that taking intoaccount the (small) polydispersity of the sample under investigation is necessary to providea good description of the dielectric normal mode contribution by means of the Rouse model,without any adjustable parameter other than the Rouse time, which (for the average molecularmass) is essentially determined from the reciprocal of the maximum loss angular frequency.

4.2.6 BDS and rheology in the same experiment

Once we have found that the Rouse model can account accurately for the chain dynamicsas observed by dielectric spectroscopy, the question that arise is if using the very sameapproach it would be possible to account also for other independent experiments, namelyrheology. A key point to perform such a test is to be sure that all the environmental sampleconditions remains the same. To be sure about this, we performed simultaneous dielectricand rheology experiments at 230 K. The sample thickness was smaller to balance the dataquality of both dielectric and rheological results. We found that a thickness of 0.9 mm usingtwo parallel plates of 25 mm diameter provides a rather good compromise. The output ofthese experiments at 230 K is shown in Figure 12. Despite of the very different geometryused, the rheological results are in close agreement with those obtained before. In thissimultaneous experiment, the dielectric normal mode is clearly resolved so the peak positiondefining the time scale can be determined with low uncertainty, although again the accuracyof the dielectric relaxation data is not so good as that obtained in the data presented before.


www.intechopen.com


Fig. 11. Resolved normal mode relaxation of PI-3 sample. The lines represent the behaviourpredicted by the Rouse model including for the actual sample polydispersity (solid line) andwithout it (dotted line). Triangles correspond to the difference between the experimentallosses and the Rouse model predictions. The inset shows schematically how the presence ofconfiguration defects allows fluctuations of the whole chain dipole moment withoutvariation of the end-to-end vector. Reprinted with permsision from Riedel et al,Macromolecules 42, 8492-8499 Copyright 2009 American Chemical Society

Nevertheless, this experiment will be essential in testing the ability of the Rouse model inaccounting simultaneously for both the dielectric and rheology signatures of the whole chaindynamics. The approach used was to determine τ1 from the dielectric losses, accordingwith the description used above that is able to accurately account for the complete dielectricrelaxation spectrum, and to use a similar approach to generate the corresponding rheologybehavior. Thus, the only unknown parameter needed to perform the comparison with theexperimental G’(ω) and G”(ω) data will be G∞ (the high frequency limit of the modulusin terminal zone), which in fact it is not needed for calculating tan (δ)= G”(ω)/G’(ω). Theequations used were obtained from Eq. RouseRheo1 and RouseRheo2 following the sameprocedure that the one from BDS:

G′(ω) =∫

G∞

Mn

N−1

∑p=1

ω2τ2p /4

1 + ω2τ2p /4

g(M)dM (29)

G′′(ω) =∫

G∞

Mn

N−1

∑p=1

ωτp/2

1 + ω2τ2p /4

g(M)dM (30)

The results of the comparison between the calculated responses and the experimental dataare shown in Fig. 12. A satisfactory description is obtained for G’(ω) and G”(ω) by means

18 Rheology

www.intechopen.com


Spectroscopy 17

Fig. 12. Simultaneous rheology (G’: �, G”: •, tan(δ) : �) and BDS experiments (ǫ”: ◦) on PI-3at 230 K. Lines correspond to the description obtained using the Rouse model, with a singleset of parameters, for all the data sets.

of the same Rouse model parameters used in describing the dielectric normal mode. Moreinterestingly, the description of tan(δ) is also rather good, for which the previous calculationis compared with the experimental data without any arbitrary scaling. Despite the goodagreement obtained, it is worthy of remark that the ability of rheology experiments forchecking the Rouse model in full detail is much more limited than the dielectric one becauseboth the narrower frequency range accessible and the stronger overlapping of the segmentaldynamics contributions.

4.3 Discussion

Despite the overall good agreement, Figure 11 evidences that the experimental losses areslightly larger (maximum difference about 10%) than the Rouse model prediction in the highfrequency side. Although one could consider that this is simply due to the contribution of theoverlapping α-relaxation that has not been properly subtracted, the fact that the maximum ofthese extra losses intensity occurs at frequencies two decades above the segmental relaxationpeak (see triangle in Figure 11) seems to point out to other origin. In agreement with this idea,that the frequency distance above the NM peak where the Rouse model description startsunderestimating the experimental losses does not depend much on temperature. Thus, extracontributions from some chain-modes are detected in the dielectric normal mode. In fact thepeak intensity of the extra losses occurs at the position of the p=3 mode (see triangles in Fig.11), i.e. the second mode contributing to the dielectric normal mode. All this might evidencethe fact that the Rouse model approach ignores several aspects of the actual chain propertiesas that of the chain stiffness Brodeck et al. (2009) or the lower friction expected to occur at thechain ends Lund et al. (2009). In this context it is noticeable that the contribution from theα-relaxation extended considerably towards the frequency range where the normal mode is


www.intechopen.com


more prominent. In fact the cut-off frequency used for describing the α-relaxation in Fig. 5was close to 100 Hz, i.e, where the differences between the normal mode response and theRouse model are more pronounced. This result indicates that for most of the high-p chainmodes the segmental relaxation is not completed. This is in contrast to the assumptions ofthe Rouse model where it is considered that all the internal motions in the chain segment areso fast that their effect can be included in the effective friction coefficient. It is noteworthythat a higher frequency cut-off would be not compatible with the experimental data of thehigh molecular weight PI samples, and would produce a more prominent underestimation ofthe dielectric normal mode losses. On the contrary, a lower frequency cut-off would improveslightly the agreement between the normal mode data and the Rouse model prediction butwould imply and higher coupling between the segmental dynamics and the whole chainmotion. Small deviations from the Rouse model predictions have also been reported fromnumerical simulations, molecular dynamics calculations and detected by neutron scatteringexperiments, Brodeck et al. (2009); Doxastakis et al. (2003); Logotheti & Theodorou (2007);Richter et al. (1999) although these deviations become evident only for relatively highp-values. Note, that dielectric experiments being mainly sensitive to the low-p modes canhardly detect such deviations. On the other hand, experiments in solution have also evidenceddifferences between the experimental data and the predictions of the Rouse model,Watanabeet al. (1995) which were tentatively attributed to chain overlapping effects since the deviationsoccur above a given concentration. Nonetheless, there are also possible experimental sourcesfor the small extra high frequency contributions to the dielectric losses as it would be thepresence in the actual polymer of a fraction of monomeric units others than the 1,4-cis ones (upto 20%). The motion of such units, having a much smaller component of the dipole momentparallel to the chain contour, would produce small amplitude fluctuations of the whole dipolemoment uncorrelated with the fluctuations of the end-to-end vector (see inset in Figure 11).The chain motions around these ’configuration defects’ would generate a relatively weakand fast contribution to the dielectric normal mode that could explain the experimental data.Unfortunately, the actual sample microstructure prevents to definitively address whether thedeviations from the Rouse model predictions are actually indicative of its limitations.

5. Dynamics regimes as a function of the molecular weight and effects of

entanglement

As shown in the previous section, the polymer dynamics follows different regimes asa function of the molecular weight and molecular weight distribution. Abdel-Goad etal Abdel-Goad et al. (2004), using rheology measurements coupled with an empiricalWinter-relaxation BSW-model obtained three different exponents (1, 3.4 and 3) in themolecular weight dependence of the zero shear viscosity. Previous BDS studies on themolecular weight dependence of the normal mode relaxation time showed a crossover fromthe unentangled dynamics to the entanglement regime Adachi & Kotak (1993); Boese &Kremer (1990); Hiroshi (2001;?); Yasuo et al. (1988). The reptational tube model was firstintroduced by de Gennes de Gennes (1979) and developed by Doi and Edwads Doi &Edwards (1988) to described this phenomena of entanglement. Many corrections (as ContourLength Fluctuation or Constraints Release) have been added to the pure reptation in orderto reach a totally predictive theory McLeish (2002); Viovy et al. (2002). However, none ofBDS experiment has been able to access the crossover to exponent 3 expected by the purereptation theory which, as aforementioned, has been detected for the viscosity. In this section,we will detail how careful BDS experiments allow detecting the two different crossovers from

20 Rheology

www.intechopen.com


Spectroscopy 19

the Rouse up to the pure reptation regime. Entanglement effects in BDS spectra will be alsoanalyzed. Finally, the possible influence of the narrow molecular weight distribution of thesamples on the dielectric loss shape will be discussed.

5.1 Low molecular weight

As can be seen in Fig. 3 the fast variation of the normal mode relaxation peak preventsdetecting it in the frequency window at 250K for molecular weight higher than 80 kg/mol.Thus, we will first focus the analysis on the samples with lower molecular weight. Asalready mentioned, for low molecular masses the changes in the local friction coefficient(arising because the significant changes in the end chain groups accompanying the changes inmolecular weight) influences the relaxation time. Therefore, just as in Fig. 8, the ratio of thenormal mode time scale to that of the α-relaxation was evaluated (Fig. 13). To increase the plot

Fig. 13. Longest and segmental relaxation time ratio as a function of the molecular weight at250 K. The vertical axis is scaled by M2 to emphasize the transition from the Rouse to theintermediate regime. The solid line corresponds to the description of the data with a sharpcrossover between two power law regimes with different exponents. Reprinted withpermsision from Riedel et al, Rheologica Acta 49, 507-512 Copyright 2010 Springer

sensitivity to changes between the different regimes a factor of M−2w has been applied to the

ratio between the characteristic times (reciprocal of the peak angular frequency) of the tworelaxation processes. As already mentioned, the M2

w dependence expected for unentangledpolymers on the basis of the Rouse theory is fulfilled for samples with molecular weightbelow 7 kg/mol. The exponent describing the higher molecular mass range considered inthis plot was 3.20.1 which is distinctly lower, but close, to the 3.4 usually found in rheologicalexperiments Ferry (1995). It is noteworthy that imposing this exponent to fit our data willresult in a higher value of the crossover molecular weight, thus increasing the discrepancywith the reported/admitted value of Me=5kg/mol measured by neutron scattering Fetterset al. (1994). It is noteworthy that, as it is well know, the ratio τN/τα will change withtemperature Ding & Sokolov (2006). Nevertheless, the previous results will not change


www.intechopen.com


significantly using data at other temperatures because the changes in the value τN/τα willbe very similar for all the samples having different molecular weights and, consequently, theresulting molecular weight dependence would be unaltered.

5.2 High molecular weight

After analyzing the molecular weight dependence of the end-to-end fluctuations in thelow and moderate molecular weight range, now we will focus the attention in the highestaccessible molecular weights. In the high molecular weight range, the comparison amongthe different samples has to be performed at a significantly higher temperature due to thedramatic slowing down of the chain dynamics. The more suitable temperatures are thosewhere the normal mode loss peak of the sample with the highest molecular weight occursin the low frequency range of the experimental window. An additional factor that have tobe taken into account is the fact that by increasing temperature the conductivity contributionto the dielectric losses becomes more prominent. The conductivity contribution appears asa ω−1 increasing of the dielectric losses. This is an important issue even for high qualitysamples when the experiments require accessing to the low frequencies at temperatures farabove Tg. This situation is illustrated in Fig. 14 for the raw data of the PI sample havingthe highest investigated molecular weight. It is apparent that at 340 K the normal mode

Fig. 14. Normal mode of the high molecular weight PI samples. Dashed line represents thecalculated conductivity contribution to the dielectric losses for the PI-320 sample. Reprintedwith permsision from Riedel et al, Rheologica Acta 49, 507-512 Copyright 2010 Springer

peak can be well resolved from conductivity for this sample but it would be hard to resolvethe normal mode relaxation at this temperature for a sample with a significantly highermolecular weight. Furthermore, increasing temperature would not improve the situationsince the overlapping of the conductivity contribution with the normal mode relaxation willalso increase. This is a serious limitation of the dielectric methods for investigating the slowestchain dynamics in highly entangled systems. Nevertheless, as shown in Fig. 14, resolving thenormal mode peak was possible for all the samples investigated although the contributionfrom conductivity will increase the uncertainty in the peak position for the samples with very

22 Rheology

www.intechopen.com


Spectroscopy 21

high molecular weight. Figure 15 shows the molecular weight dependence of the slowestrelaxation time for the high molecular weight regime obtained from the data presented inFig. 14. Trying to increase the sensitivity of the plot to possible changes in behavior the data

Fig. 15. Longest relaxation time from the higher molecular weight PI samples. The graph isscaled to M3 to emphasize the crossover from the intermediate to the reptation regime. Thesolid line corresponds to the description of the data with a sharp crossover between to powerlaw regimes with different exponents. Reprinted with permsision from Riedel et al,Rheologica Acta 49, 507-512 Copyright 2010 Springer

have been multiplied by M−3w , which would produce a molecular weight independent result

for a pure reptation regime. Despite of the uncertainties involved, our results evidence thatfor the highest molecular weight samples the molecular mass dependence approach the purereptation regime expectation. The line in Fig. 4 corresponds to a crossover from and exponent3.35 to a pure reptation-like regime. The small difference between this exponent and thatobtained above from Figure 2 is more likely due to the fact that the sample with the lowermolecular weight considered in Fig. 4 have a significantly lower glass transition temperature,an effect not been considered in Fig. 4. The crossover molecular weight obtained from Fig. 4is 7510 kg/mol, i.e. it corresponds to about 15 times Me. This value is slightly lower than thatdetermined from viscosity data Abdel-Goad et al. (2004).

5.3 Discussion

The results previously described showed three different regimes for the molecular weightdependence of the chain longest relaxation time in PI, one below 7 kg/mol following theRouse model prediction as expected for a non-entangled polymer melts, other above 75kg/mol where the reptation theory provides a good description and an intermediate one,where the polymer is entangled but other mechanisms (like contour length fluctuations orconstraints release) in addition to reptation would control the whole chain dynamics. Inthe rheological experiments above referred Abdel-Goad et al. (2004) it was shown that theviscosity of high molecular weight PI samples conforms well the reptation theory predictions.


www.intechopen.com


Thus, we decided to test up to what extent the pure reptation theory is able to describe thenormal mode relaxation spectrum of the highly entangled PI samples. This test can evidencethe ability of the reptation theory to capture the main features of the slowest chain dynamics,despite the well documented failure of the reptation theory in accounting for the whole chaindynamics, even in the high molecular weight range. This is clearly evidenced by the reportedmismatching of the normalized dielectric and rheological spectra Watanabe et al. (2002b).To this end, we compared our experimental data on the high molecular mass samples withthe corresponding reptation theory predictions for the dielectric permittivity. This relation issimilar to the one obtained in the frame of the Rouse model (Eq. 23. As the chain are long, N>pand the cot2 can be developed in 1/p2. The value of τp = τ1/p2 is replaced by τd/p2 where τd

is the disentanglement time (reptation) time, which would correspond in good approximationto τN .

ǫ′′(ω) ∝N−1

∑p:odd

1

p2

ω τd/p2

1 + (ω2 τd/p2)2(31)

Figure 16 shows the direct comparison between the experimental data for some of the samplesinvestigated (symbols) having all of them the lowest available polydispersity index ( 1.05) andthe pure reptation theory prediction (solid line). Both vertical and horizontal scaling factorshave been applied to obtain a good matching of the peaks. It should be noted that the possibleconductivity contributions to the normal mode relaxation were subtracted. The inset showsseparately the data of the highest molecular weight sample because it has a markedly broadermolecular weight distribution (polydispersity index 1.14). From Fig. 16 it becomes apparent

Fig. 16. Comparison of the BDS data of high molecular weight (symbols) with pure reptationtheory (solid line). Dashed straight line showing the ij power law behavior of the sampleswith PI-33 and PI-145 are also shown. The vertical arrows indicate the crossover frequencybetween both regimes. The inset presents for comparison the highest molecular weight data(PI-320) with a high polydispersity (1.14) with that of smaller polydispersity (1.04). Thedashed line represents what would be the reptation theory expectation when a very crudeapproximation is used to account the effect of the molecular weight. Reprinted withpermsision from Riedel et al, Rheologica Acta 49, 507-512 Copyright 2010 Springer

that in the high frequency side of the loss peak deviations from the reptation predictions on

24 Rheology

www.intechopen.com


Spectroscopy 23

the end-to-end vector fluctuations persist even for the highest molecular weight investigated.Whereas the high frequency behavior expected from the reptation theory is a power lawwith exponent -1/2, the experimental data present and exponent -1/4 (see Figure 16), whichwould be a signature of the relevance of chain contour length fluctuations at least in this highfrequency side of the normal mode relaxation. Nevertheless, it is also clear that the rangeof these deviations reduce when increasing molecular weight. The vertical arrows in Fig.5 shows that a factor of 5 increasing in molecular weight makes the crossover frequency toincrease in a factor of about 2. By inspection of the rheological data reported by Abdel-Goadet al, Abdel-Goad et al. (2004) it is also apparent that in the very high molecular weightrange where the viscosity scales as predicted by reptation theory the terminal relaxation isfar from being properly described by this theory. Eventually the normal mode descriptionby the pure reptation theory could be obtained only at extremely higher molecular weights,for which, as aforementioned, the dielectric experiments will not be suitable for investigatingthe extremely slow chain dynamics. Concerning this, it has been shown McLeish (2002) thatfor polyethylene de frequency dependence of the loss shear modulus verifies the reptationprediction only for a molecular weight as high as of 800 kg/mol, which for this polymercorresponds to about 400 times Me, i.e., it would correspond to about 3000 kg/mol for PI.Taking the above-calculated shift of the crossover frequency into account, for this limitingmolecular weight the crossover frequency would occur at around 20 Hz and the failure ofthe repetition theory description would be hardly detectable by using the same scale asin Figure 16. Figure 16 also shows that both the maximum and the low frequency sideof the loss peak is well accounted by the reptation theory without any evident deviation,except for the sample having a broader distribution of molecular weight, which shows adistinctly broader normal mode peak (see inset of Fig. 16). This comparison evidences thatthe molecular weight distribution have a noticeable effect on the normal mode spectrumshape. We remind that the effect of the molecular weight distribution on the normal modewas properly accounted for in an unentangled PI sample by assuming that the contributionsfrom chains in the sample with distinct molecular weight simply superimpose (see previoussection on the Rouse model). When we tried the same approach with the higher molecularweight samples (dashed line in the inset of Fig. 16) it becomes evident that the situationfor well-entangled polymers is different. Even by using the smallest polydispersity (1.04)the calculated response overestimates by far the broadening of the peak for the sample withhighest polydispersity (1.14). Thus, for highly entangled polymers the effect of the molecularweight distribution on the normal mode is less evident than that observed in the unentangledpolymer case. In fact, the complete disentanglement of a chain involves also the motions of thechains around, which would have a different molecular weight, being therefore the resultingtime scale some kind of average of those corresponding to the ideally monodisperse melts. Asa result the longest relaxation time in highly entangled melts should not dependent greatly onthe molecular weight distribution provided it is not very broad.

6. Conclusion

In this chapter, we have used broadband dielectric spectroscopy (BDS) and rheology to studyproperties of linear polymers. We have focused our study on the large chain dynamics (normalmode), described by the reptational tube theory and the Rouse model when the polymer isentangled or not, respectively.


www.intechopen.com


By means of broad-band dielectric spectroscopy we have resolved the normal mode relaxationof a non-entangled 1,4-cis-poly(isoprene) (PI) and therefore accessed to the end-to-end vectordynamics over a broad frequency/time range. A remarkably good comparison of the datawith the Rouse model predictions is found if the effect of the actual narrow distribution ofmolecular masses of the sample investigated is accounted. The very same approach wasfound to provide a good description of a simultaneous dielectric/rheology experiment. Thesmall excess contributions found in the high frequency side of the experimental dielectricnormal mode losses could be associated, at least partially, to the sample microstructure details.Therefore, we conclude that the Rouse model accounts within experimental uncertainties forthe end-to-end dynamics of unentangled PI, once the molecular weight distribution effectsare considered. A more sensitive test would require an unentangled nearly monodisperse full1,4-cis-polyisoprene sample, which can hardly be available.

Further experiments have permitted to detect two crossovers in the molecular weightdependence of the end-to-end relaxation time. The first corresponds to the crossover fromthe range where the Rouse theory is applicable to the entangled limited range, being thecrossover molecular weight 6.5 ± 0.5 kg/mol, i.e. slightly above the molecular weightbetween entanglements. The crossover from the intermediate range to the behavior predictedby the pure reptation theory is found at around 75 ± 10 kg/mol, which corresponds to 15times the molecular weight between entanglements. Despite of the fact that the reptationtheory is able to describe the molecular weight dependence of the slowest relaxation timefor these high molecular weight samples, the shape of the normal mode spectrum is stillmarkedly different from that expected by this theory. Eventually, only at a much highermolecular weight (hundred times the molecular weight between entanglements) the reptationtheory could completely describe the normal mode relaxation associate to the chain dynamics.Unfortunately, dielectric experiments in this range are not feasible.

To conclude, in this chapter we have shown that dielectric spectroscopy and rheologypermitted to obtain physical information about polymer. Even if these two techniquesmeasure different properties, the origin of these properties, is the same: the dynamics. Asthey permit to extend our knowledge on polymer dynamics, these techniques will allow abetter understanding and control of polymer properties.

7. References

Abdel-Goad, M., Pyckhout-Hintzen, W., Kahle, S., Allgaier, J., Richter, D. & Fetters, L. J.(2004). Rheological properties of 1,4-polyisoprene over a large molecular weightrange, Macromolecules 37(21): 8135–8144.

Adachi, K. & Kotak (1993). Dielectric normal mode relaxation, Progress in polymer science18(3): 585.

Adachi, K., Yoshida, H., Fukui, F. & Kotaka, T. (1990). Comparison of dielectric andviscoelastic relaxation spectra of polyisoprene, Macromolecules 23(12): 3138–3144. doi:10.1021/ma00214a018.

Boese, D. & Kremer, F. (1990). Molecular dynamics in bulk cis-polyisoprene as studied bydielectric spectroscopy, Macromolecules 23(3): 829–835.

Brodeck, M., Alvarez, F., Arbe, A., Juranyi, F., Unruh, T., Holderer, O., Colmenero, J. & Richter,D. (2009). Study of the dynamics of poly(ethylene oxide) by combining moleculardynamic simulations and neutron scattering experiments, The Journal of ChemicalPhysics 130(9): 094908.

26 Rheology

www.intechopen.com


Spectroscopy 25

Cangialosi, D., Alegria, A. & Colmenero, J. (2007). Route to calculate the length scale for theglass transition in polymers, Physical Review E 76(1): 011514. Copyright (C) 2010 TheAmerican Physical Society Please report any problems to [email protected] PRE.

de Gennes, P. G. (1979). Scaling Concepts in Polymer Physics, Cornell University Press, Ithacaand London.

Ding, Y. & Sokolov, A. P. (2006). Breakdown of time temperature superposition principle anduniversality of chain dynamics in polymers, Macromolecules 39(9): 3322–3326.

Doi, M. & Edwards, S. F. (1988). The theory of polymer dynamics, Clarendon, Oxford.Doxastakis, M., Theodorou, D. N., Fytas, G., Kremer, F., Faller, R., Muller-Plathe, F.

& Hadjichristidis, N. (2003). Chain and local dynamics of polyisoprene asprobed by experiments and computer simulations, The Journal of Chemical Physics119(13): 6883–6894.

Ferry, J. D. (1995). Viscoelastic Properties of Polymers, Oxford University Press, New York.Fetters, L. J., Lohse, D. J., Richter, D., Witten, T. A. & Zirkel, A. (1994). Connection

between polymer molecular weight, density, chain dimensions, and melt viscoelasticproperties, Macromolecules 27(17): 4639–4647. doi: 10.1021/ma00095a001.

Hiroshi, W. (2001). Dielectric relaxation of type-a polymers in melts and solutions,Macromolecular Rapid Communications 22(3): 127–175. 10.1002/1521-3927(200102) 22:3<127::AID-MARC127>3.0.CO;2-S.

Inoue, T., Uematsu, T. & Osaki, K. (2002a). The significance of the rouse segment: Itsconcentration dependence, Macromolecules 35(3): 820–826. doi: 10.1021/ma011037m.

Inoue, T., Uematsu, T. & Osaki, K. (2002b). The significance of the rouse segment: Itsconcentration dependence, Macromolecules 35(3): 820–826. doi: 10.1021/ma011037m.

Kremer, F. & Schonals, A. (2003). Broadband Dielectric Spectroscopy, Springer, Berlin.Likhtman, A. E. & McLeish, T. C. B. (2002). Quantitative theory for linear dynamics of linear

entangled polymers, Macromolecules 35(16): 6332–6343. doi: 10.1021/ma0200219.Lodge, T. P. & McLeish, T. C. B. (2000). Self-concentrations and effective glass transition

temperatures in polymer blends, Macromolecules 33(14): 5278–5284. doi: 10.1021/ma9921706.

Logotheti, G. E. & Theodorou, D. N. (2007). Segmental and chain dynamics of isotacticpolypropylene melts, Macromolecules 40(6): 2235–2245. doi: 10.1021/ma062234u.

Lund, R., Plaza-Garcia, S., Alegria, A., Colmenero, J., Janoski, J., Chowdhury, S. R. &Quirk, R. P. (2009). Polymer dynamics of well-defined, chain-end-functionalizedpolystyrenes by dielectric spectroscopy, Macromolecules 42(22): 8875–8881. doi:10.1021/ma901617u.

McLeish, T. C. B. (2002). Tube theory of entangled polymer dynamics, Advances in Physics51(6): 1379.

Richter, D., M.Monkenbusch, Arbe, A. & Colmenero, J. (2005). Neutron Spin Echo in PolymerSystems, Springer, Berlin.

Richter, D., Monkenbusch, M., Allgeier, J., Arbe, A., Colmenero, J., Farago, B., Bae, Y. C. &Faust, R. (1999). From rouse dynamics to local relaxation: A neutron spin echo studyon polyisobutylene melts, The Journal of Chemical Physics 111(13): 6107–6120.

Rouse, P. E. (1953). A theory for the linear elasticity properties of dilute solutions of coilingpolymers, The Journal of Chemical Physics 21(7).

Rubinstein, M. & Colby, R. H. (2003). Polymer Physics, Oxford University, New York.Stockmayer, W. (1967). Dielectric disperson in solutions of flexible polymers, Pure Applied

Chemesitry p. 539.


www.intechopen.com


Viovy, J. L., Rubinstein, M. & Colby, R. H. (2002). Constraint release in polymer melts:tube reorganization versus tube dilation, Macromolecules 24(12): 3587–3596. doi:10.1021/ma00012a020.

Watanabe, H., Matsumiya, Y. & Inoue, T. (2002a). Dielectric and viscoelastic relaxationof highly entangled star polyisoprene: Quantitative test of tube dilation model,Macromolecules 35(6): 2339–2357. doi: 10.1021/ma011782z.

Watanabe, H., Matsumiya, Y. & Inoue, T. (2002b). Dielectric and viscoelastic relaxationof highly entangled star polyisoprene: Quantitative test of tube dilation model,Macromolecules 35(6): 2339–2357. doi: 10.1021/ma011782z.

Watanabe, H., Sawada, T. & Matsumiya, Y. (n.d.). Constraint release in star/star blends andpartial tube dilation in monodisperse star systems, Macromolecules 39(7): 2553–2561.doi: 10.1021/ma0600198.

Watanabe, H., Yamada, H. & Urakawa, O. (1995). Dielectric relaxation of dipole-invertedcis-polyisoprene solutions, Macromolecules 28(19): 6443–6453. doi: 10.1021/ma00123a009.

Williams, M. L., Landel, R. F. & Ferry, J. D. (1995). Mechanical properties of substances of highmolecular weight in amorphous polymers and other glass-forming liquids, Journal ofAmerican Chemical Society 77(19): 3701–3707.

Yasuo, I., Keiichiro, A. & Tadao, K. (1988). Further investigation of the dielectric normal modeprocess in undiluted cis-polyisoprene with narrow distribution of molecular weight,The Journal of Chemical Physics 89(12): 7585–7592.

28 Rheology

www.intechopen.com

RheologyEdited by Dr. Juan De Vicente

ISBN 978-953-51-0187-1Hard cover, 350 pagesPublisher InTechPublished online 07, March, 2012Published in print edition March, 2012

InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.com

InTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China

Phone: +86-21-62489820 Fax: +86-21-62489821

This book contains a wealth of useful information on current rheology research. By covering a broad variety ofrheology-related topics, this e-book is addressed to a wide spectrum of academic and applied researchers andscientists but it could also prove useful to industry specialists. The subject areas include, polymer gels, foodrheology, drilling fluids and liquid crystals among others.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Clement Riedel, Angel Alegria, Juan Colmenero and Phillipe Tordjeman (2012). Polymer Rheology byDielectric Spectroscopy, Rheology, Dr. Juan De Vicente (Ed.), ISBN: 978-953-51-0187-1, InTech, Availablefrom: http://www.intechopen.com/books/rheology/polymer-rheology-by-dielectric-spectroscopy

© 2012 The Author(s). Licensee IntechOpen. This is an open access articledistributed under the terms of the Creative Commons Attribution 3.0License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

http://creativecommons.org/licenses/by/3.0

Documents

Polymer Rheology by Dielectric Spectroscopy · 2018. 9. 25. · 0 Polymer Rheology by Dielectric Spectroscopy Clement Riedel 1, Angel Alegria 1, Juan Colmenero 1 and Phillipe Tordjeman